Fundamentals of
Power Electronics
SECOND EDITION
Robert W. Erickson
Dragan Maksimovic
University of Colorado
Boulder, Colorado
Contents
Preface
xix
1
Introduction
1.1
Introduction to Power Processing
1.2
Several Applications of Power Electronics
1.3
Elements of Power Electronics
References
1
Converters in Equilibrium
11
2
Principles of Steady State Converter Analysis
13
2.1
2.2
13
Introduction
Inductor Volt-Second Balance, Capacitor Charge Balance, and the Small-Ripple
Approximation
2.3
Boost Converter Example
2.4
Cuk Converter Example
2.5
Estimating the Output Voltage Ripple in Converters Containing Two-Pole
Low-Pass Filters
2.6
Summary of Key Points
References
Problems
3
1
1
7
9
15
22
27
31
34
34
35
Steady-State Equivalent Circuit Modeling, Losses, and Efficiency
39
3.1
3.2
3.3
39
42
45
The DC Transformer Model
Inclusion of Inductor Copper Loss
Construction of Equivalent Circuit Model
viii
4
5
6
Contents
3.3.1 Inductor Voltage Equation
3.3.2 Capacitor Current Equation
3.3.3 Complete Circuit Model
3.3.4 Efficiency
3.4 How to Obtain the Input Port of the Model
3.5 Example: Inclusion of Semiconductor Conduction Losses in the Boost
Converter Model
3.6 Summary of Key Points
References
Problems
46
46
47
48
50
52
56
56
57
Switch Realization
63
4.1
Switch Applications
4.1.1 Single-Quadrant Switches
4.1.2 Current-Bidirectional Two-Quadrant Switches
4.1.3 Voltage-Bidirectional Two-Quadrant Switches
4.1.4
Four-Quadrant Switches
4.1.5 Synchronous Rectifiers
4.2
A Brief Survey of Power Semiconductor Devices
4.2.1 Power Diodes
4.2.2 Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET)
4.2.3 Bipolar Junction Transistor (BJT)
4.2.4 Insulated Gate Bipolar Transistor (IGBT)
4.2.5 Thyristors (SCR, GTO, MCT)
4.3 Switching Loss
4.3.1 Transistor Switching with Clamped Inductive Load
4.3.2 Diode Recovered Charge
4.3.3 Device Capacitances, and Leakage, Package, and Stray Inductances
4.3.4 Efficiency vs. Switching Frequency
4.4 Summary of Key Points
References
Problems
65
65
67
71
72
73
74
75
78
81
86
88
92
93
96
98
100
101
102
103
The Discontinuous Conduction Mode
107
5.1 Origin of the Discontinuous Conduction Mode, and Mode Boundary
5.2 Analysis of the Conversion Ratio Мф,К)
5.3 Boost Converter Example
5.4 Summary of Results and Key Points
Problems
108
112
117
124
126
Converter Circuits
131
6.1
132
132
134
137
Circuit Manipulations
6.1.1
Inversion of Source and Load
6.1.2 Cascade Connection of Converters
6.1.3
Rotation of Three-Terminal Cell
Contents
ix
6.1.4
Differential Connection of the Load
6.2
A Short List of Converters
6.3
Transformer Isolation
6.3.1
Full-Bridge and Half-Bridge Isolated Buck Converters
6.3.2
Forward Converter
6.3.3
Push-Pull Isolated Buck Converter
6.3.4
Flyback Converter
6.3.5
Boost-Derived Isolated Converters
6.3.6
Isolated Versions of the SEPIC and the Cuk Converter
6.4
Converter Evaluation and Design
6.4.1
Switch Stress and Utilization
6.4.2
Design Using Computer Spreadsheet
6.5
Summary of Key Points
References
Problems
138
143
146
149
154
159
161
165
168
171
171
174
177
177
179
Converter Dynamics and Control
185
AC Equivalent Circuit Modeling
187
7.1
7.2
Introduction
The Basic AC Modeling Approach
187
192
7.2.1
Averaging the Inductor Waveforms
7.2.2
Discussion of the Averaging Approximation
7.2.3
Averaging the Capacitor Waveforms
7.2.4
The Average Input Current
7.2.5
Perturbation and Linearization
7.2.6
Construction of the Small-Signal Equivalent Circuit Model
7.2.7
Discussion of the Perturbation and Linearization Step
7.2.8
Results for Several Basic Converters
7.2.9
Example: A Nonideal Flyback Converter
State-Space Averaging
7.3.1
The State Equations of a Network
7.3.2
The Basic State-Space Averaged Model
7.3.3
Discussion of the State-Space Averaging Result
7.3.4
Example: State-Space Averaging of a Nonideal Buck-Boost Converter
Circuit Averaging and Averaged Switch Modeling
7.4.1
Obtaining a Time-Invariant Circuit
7.4.2
Circuit Averaging
7.4.3
Perturbation and Linearization
7.4.4
Switch Networks
7.4.5
Example: Averaged Switch Modeling of Conduction Losses
7.4.6
Example: Averaged Switch Modeling of Switching Losses
The Canonical Circuit Model
7.5.1
Development of the Canonical Circuit Model
193
194
196
197
197
201
202
204
204
213
213
216
217
221
226
228
229
232
235
242
244
247
248
7.3
7.4
7.5
x
Contents
7.5.2
8
9
Example: Manipulation of the Buck-Boost Converter Model
into Canonical Form
7.5.3 Canonical Circuit Parameter Values for Some Common Converters
7.6
Modeling the Pulse-Width Modulator
7.7
Summary of Key Points
References
Problems
250
252
253
256
257
258
Converter Transfer Functions
265
8.1
Review of Bode Plots
8.1.1 Single Pole Response
8.1.2 Single Zero Response
8.1.3 Right Half-Plane Zero
8.1.4 Frequency Inversion
8.1.5
Combinations
8.1.6 Quadratic Pole Response: Resonance
8.1.7 The Low-Q Approximation
8.1.8 Approximate Roots of an Arbitrary-Degree Polynomial
8.2 Analysis of Converter Transfer Functions
8.2.1 Example: Transfer Functions of the Buck-Boost Converter
8.2.2 Transfer Functions of Some Basic CCM Converters
8.2.3 Physical Origins of the RHP Zero in Converters
8.3 Graphical Construction of Impedances and Transfer Functions
8.3.1 Series Impedances: Addition of Asymptotes
8.3.2 Series Resonant Circuit Example
8.3.3 Parallel Impedances: Inverse Addition of Asymptotes
8.3.4
Parallel Resonant Circuit Example
8.3.5 Voltage Divider Transfer Functions: Division of Asymptotes
8.4 Graphical Construction of Converter Transfer Functions
8.5 Measurement of AC Transfer Functions and Impedances
8.6 Summary of Key Points
References
Problems
267
269
275
276
277
278
282
287
289
293
294
300
300
302
303
305
308
309
311
313
317
321
322
322
Controller Design
331
9.1
9.2
331
334
9.3
9.4
Introduction
Effect of Negative Feedback on the Network Transfer Functions
9.2.1 Feedback Reduces the Transfer Functions
from Disturbances to the Output
9.2.2 Feedback Causes the Transfer Function from the Reference Input
to the Output to be Insensitive to Variations in the Gains in the
Forward Path of the Loop
Construction of the Important Quantities 1/(1 + T) and 77(1 + T)
and the Closed-Loop Transfer Functions
Stability
335
337
337
340
Contents
9.4.1
9.4.2
xi
The Phase Margin Test
The Relationship Between Phase Margin
and Closed-Loop Damping Factor
9.4.3 Transient Response vs. Damping Factor
9.5 Regulator Design
9.5.1 Lead (PD) Compensator
9.5.2 Lag (PI) Compensator
9.5.3 Combined (PID) Compensator
9.5.4 Design Example
9.6
Measurement of Loop Gains
9.6.1 Voltage Injection
9.6.2 Current Injection
9.6.3 Measurement of Unstable Systems
9.7 Summary of Key Points
References
Problems
341
342
346
347
348
351
353
354
362
364
367
368
369
369
369
Input Filter Design
377
10.1
Introduction
10.1.1 Conducted EMI
10.1.2 The Input Filter Design Problem
10.2 Effect of an Input Filter on Converter Transfer Functions
10.2.1 Discussion
10.2.2 Impedance Inequalities
10.3 Buck Converter Example
10.3.1 Effect of Undamped Input Filter
10.3.2 Damping the Input Filter
10.4 Design of a Damped Input Filter
10.4.1 Rf-Cb Parallel Damping
10.4.2 Rf-Lb Parallel Damping
10.4.3 Rf-Lb Series Damping
10.4.4 Cascading Filter Sections
10.4.5 Example: Two Stage Input Filter
10.5 Summary of Key Points
References
Problems
AC and DC Equivalent Circuit Modeling of the Discontinuous Conduction Mode
377
377
379
381
382
384
385
385
391
392
395
396
398
398
400
403
405
406
409
11.1 DCM Averaged Switch Model
11.2 Small-Signal AC Modeling of the DCM Switch Network
11.2.1 Example: Control-to-Output Frequency Response
of a DCM Boost Converter
11.2.2 Example: Control-to-Output Frequency Responses
ofaCCM/DCMSEPIC
410
420
428
429
xü
12
Contents
11.3 High-Frequency Dynamics of Converters in DCM
11.4 Summary of Key Points
References
Problems
431
434
434
435
Current Programmed Control
439
12.1 Oscillation for D> 0.5
12.2 A Simple First-Order Model
12.2.1 Simple Model via Algebraic Approach: Buck-Boost Example
12.2.2 Averaged Switch Modeling
12.3 A More Accurate Model
12.3.1 Current-Programmed Controller Model
12.3.2 Solution of the CPM Transfer Functions
12.3.3 Discussion
12.3.4 Current-Programmed Transfer Functions of the CCM Buck Converter
12.3.5 Results for Basic Converters
12.3.6 Quantitative Effects of Current-Programmed Control
on the Converter Transfer Functions
12.4 Discontinuous Conduction Mode
12.5 Summary of Key Points
References
Problems
441
449
450
454
459
459
462
465
466
469
471
473
480
481
482
III Magnetics
489
13 Basic Magnetics Theory
491
13.1 Review of Basic Magnetics
13.1.1 Basic Relationships
13.1.2 Magnetic Circuits
13.2 Transformer Modeling
13.2.1 The Ideal Transformer
13.2.2 The Magnetizing Inductance
13.2.3 Leakage Inductances
13.3 Loss Mechanisms in Magnetic Devices
13.3.1 Core Loss
13.3.2 Low-Frequency Copper Loss
13.4 Eddy Currents in Winding Conductors
13.4.1 Introduction to the Skin and Proximity Effects
13.4.2 Leakage Flux in Windings
13.4.3 Foil Windings and Layers
13.4.4 Power Loss in a Layer
13.4.5 Example: Power Loss in a Transformer Winding
13.4.6 Interleaving the Windings
13.4.7 PWM Waveform Harmonics
491
491
498
501
502
502
504
506
506
508
508
508
512
514
515
518
520
522
Contents
xiii
13.5
Several Types of Magnetic Devices, Their B-H Loops,
and Core vs. Copper Loss
13.5.1 Filter Inductor
13.5.2 AC Inductor
13.5.3 Transformer
13.5.4 Coupled Inductor
13.5.5 Flyback Transformer
13.6 Summary of Key Points
References
Problems
525
525
527
528
529
530
531
532
533
Inductor Design
539
14.1
Filter Inductor Design Constraints
14.1.1 Maximum Flux Density
14.1.2 Inductance
14.1.3 Winding Area
14.1.4 Winding Resistance
14.1.5 The Core Geometrical Constant К
14.2
A Step-by-Step Procedure
Multiple-Winding Magnetics Design via the К Method
539
541
542
542
543
543
544
545
14.3
14.3.1 Window Area Allocation
14.3.2 Coupled Inductor Design Constraints
14.3.3 Design Procedure
14.4 Examples
14.4.1 Coupled Inductor for a Two-Output Forward Converter
14.4.2 CCM Flyback Transformer
14.5 Summary of Key Points
References
Problems
545
550
552
554
554
557
562
562
563
Transformer Design
565
15.1
565
566
566
567
568
569
570
573
573
576
580
580
582
15.2
15.3
15.4
Transformer Design: Basic Constraints
15.1.1 Core Loss
15.1.2 Flux Density
15.1.3 Copper Loss
15.1.4 Total Power Loss vs. ДВ
15.1.5 Optimum Flux Density
A Step-by-Step Transformer Design Procedure
Examples
15.3.1 Example 1: Single-Output Isolated Cuk Converter
15.3.2 Example 2: Multiple-Output Full-Bridge Buck Converter
AC Inductor Design
15.4.1 Outline of Derivation
15.4.2 Step-by-Step AC Inductor Design Procedure
xiv
Contents
15.5 Summary
References
Problems
583
583
584
IV Modern Rectifiers and Power System Harmonics
587
16 Power and Harmonics in Nonsinusoidal Systems
589
16.1 Average Power
16.2 Root-Mean-Square (RMS) Value of a Waveform
16.3 Power Factor
16.3.1 Linear Resistive Load, Nonsinusoidal Voltage
16.3.2 Nonlinear Dynamic Load, Sinusoidal Voltage
16.4 Power Phasors in Sinusoidal Systems
16.5 Harmonic Currents in Three-Phase Systems
16.5.1 Harmonic Currents in Three-Phase Four-Wire Networks
16.5.2 Harmonic Currents in Three-Phase Three-Wire Networks
16.5.3 Harmonic Current Flow in Power Factor Correction Capacitors
16.6 AC Line Current Harmonic Standards
16.6.1 International Electrotechnical Commission Standard 1000
16.6.2 IEEE/ANSI Standard 519
Bibliography
Problems
590
593
594
594
595
598
599
599
601
602
603
603
604
605
605
17 Line-Commutated Rectifiers
609
17.1 The Single-Phase Full-Wave Rectifier
17.1.1 Continuous Conduction Mode
17.1.2 Discontinuous Conduction Mode
17.1.3 Behavior when С is Large
17.1.4 Minimizing THD when С is Small
17.2 The Three-Phase Bridge Rectifier
17.2.1 Continuous Conduction Mode
17.2.2 Discontinuous Conduction Mode
17.3 Phase Control
17.3.1 Inverter Mode
17.3.2 Harmonics and Power Factor
17.3.3 Commutation
17.4 Harmonic Trap Filters
17.5 Transformer Connections
17.6 Summary
References
Problems
18 Pulse-Width Modulated Rectifiers
609
610
611
612
613
615
615
616
617
619
619
620
622
628
630
631
632
637
18.1
Properties of the Ideal Rectifier
638
Contents
18.2 Realization of a Near-Ideal Rectifier
18.2.1 CCM Boost Converter
18.2.2 DCM Flyback Converter
18.3 Control of the Current Waveform
18.3.1 Average Current Control
18.3.2 Current Programmed Control
18.3.3 Critical Conduction Mode and Hysteretic Control
18.3.4 Nonlinear Carrier Control
18.4 Single-Phase Converter Systems Incorporating Ideal Rectifiers
18.4.1 Energy Storage
18.4.2 Modeling the Outer Low-Bandwidth Control System
18.5 RMS Values of Rectifier Waveforms
18.5.1 Boost Rectifier Example
18.5.2 Comparison of Single-Phase Rectifier Topologies
18.6 Modeling Losses and Efficiency in CCM High-Quality Rectifiers
18.6.1 Expression for Controller Duty Cycle d(t)
18.6.2 Expression for the DC Load Current
18.6.3 Solution for Converter Efficiency Ti
18.6.4 Design Example
18.7 Ideal Three-Phase Rectifiers
18.8 Summary of Key Points
References
Problems
640
642
646
648
648
654
657
659
663
663
668
673
674
676
678
679
681
683
684
685
691
692
696
V Resonant Converters
703
19 Resonant Conversion
705
19.1 Sinusoidal Analysis of Resonant Converters
19.1.1 Controlled Switch Network Model
19.1.2 Modeling the Rectifier and Capacitive Filter Networks
19.1.3 Resonant Tank Network
19.1.4 Solution of Converter Voltage Conversion Ratio M = V/V
19.2 Examples
19.2.1 Series Resonant DC-DC Converter Example
19.2.2 Subharmonic Modes of the Series Resonant Converter
19.2.3 Parallel Resonant DC-DC Converter Example
19.3 Soft Switching
19.3.1 Operation of the Full Bridge Below Resonance:
Zero-Current Switching
19.3.2 Operation of the Full Bridge Above Resonance:
Zero-Voltage Switching
19.4 Load-Dependent Properties of Resonant Converters
19.4.1 Inverter Output Characteristics
19.4.2 Dependence of Transistor Current on Load
19.4.3 Dependence of the ZVS/ZCS Boundary on Load Resistance
709
710
711
713
714
715
715
717
718
721
722
723
726
727
729
734
xvi
Contents
19.4.4 Another Example
Exact Characteristics of the Series and Parallel Resonant Converters
19.5.1 Series Resonant Converter
19.5.2 Parallel Resonant Converter
19.6 Summary of Key Points
References
Problems
737
740
740
748
752
752
755
Soft Switching
761
20.1
762
763
765
768
768
770
774
779
781
19.5
20
20.2
20.3
Soft-Switching Mechanisms of Semiconductor Devices
20.1.1 Diode Switching
20.1.2 MOSFET Switching
20.1.3 IGBT Switching
The Zero-Current-Switching Quasi-Resonant Switch Cell
20.2.1 Waveforms of the Half-Wave ZCS Quasi-Resonant Switch Cell
20.2.2 The Average Terminal Waveforms
20.2.3 The Full-Wave ZCS Quasi-Resonant Switch Cell
Resonant Switch Topologies
20.3.1 The Zero-Voltage-Switching Quasi-Resonant Switch
20.3.2 The Zero-Voltage-Switching Multi-Resonant Switch
20.3.3 Quasi-Square-Wave Resonant Switches
20.4 Soft Switching in PWM Converters
20.4.1 The Zero-Voltage Transition Full-Bridge Converter
20.4.2 The Auxiliary Switch Approach
20.4.3 Auxiliary Resonant Commutated Pole
20.5 Summary of Key Points
References
Problems
Appendices
Appendix A
783
784
787
790
791
794
796
797
798
800
803
RMS Values of Commonly-Observed Converter Waveforms
805
A.l
Some Common Waveforms
805
A.2
General Piecewise Waveform
809
Appendix В
B. 1
B.2
Simulation of Converters
Averaged Switch Models for Continuous Conduction Mode
B.l. 1 Basic CCM Averaged Switch Model
B.1.2 CCM Subcircuit Model that Includes Switch Conduction Losses
B. 1.3
Example: SEPIC DC Conversion Ratio and Efficiency
B.1.4
Example: Transient Response of a Buck-Boost Converter
Combined CCM/DCM Averaged Switch Model
B.2.1
Example: SEPIC Frequency Responses
B.2.2
Example: Loop Gain and Closed-Loop Responses
of a Buck Voltage Regulator
813
815
815
816
818
819
822
825
827
Contents
B.3
B.2.3
Current
B.3.1
B.3.2
Example: DCM Boost Rectifier
Programmed Control
Current Programmed Mode Model for Simulation
Example: Frequency Responses of a Buck Converter with
Current Programmed Control
References
Appendix С
Middlebrook's Extra Element Theorem
C.l
C.2
C.3
C.4
Basic Result
Derivation
Discussion
Examples
C.4.1
A Simple Transfer Function
C.4.2
An Unmodeled Element
C.4.3
Addition of an Input Filter to a Converter
C.4.4
Dependence of Transistor Current on Load in a Resonant Inverter
References
Appendix D
Magnetics Design Tables
D.l
Pot Core Data
D.2 ЕЕ Core Data
D.3 EC Core Data
D.4 ETD Core Data
D.5 PQ Core Data
D.6 American Wire Gauge Data
References
Index
xvii
832
834
834
837
840
843
843
846
849
850
850
855
857
859
861
863
864
865
866
866
867
868
869
871